Level One Modular Forms¶
Computing \(\Delta\)¶
The modular form
is perhaps the world’s most famous modular form. We compute some terms from the definition.
sage: R.<q> = QQ[[]]
sage: q * prod( 1-q^n+O(q^6) for n in (1..5) )^24
q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 - 6048*q^6 + O(q^7)
There are much better ways to compute \(\Delta\), which amount to just a few polynomial multiplications over \(\ZZ\).
sage: D = delta_qexp(10^5) # less than 10 seconds
sage: D[:10]
q - 24*q^2 + 252*q^3 - 1472*q^4 + ...
sage: [p for p in primes(10^5) if D[p] % p == 0]
[2, 3, 5, 7, 2411]
sage: D[2411]
4542041100095889012
sage: f = eisenstein_series_qexp(12,6) - D[:6]; f
691/65520 + 2073*q^2 + 176896*q^3 + 4197825*q^4 + 48823296*q^5 + O(q^6)
sage: f % 691
O(q^6)
The Victor Miller Basis¶
The Victor Miller basis for \(M_k(\mathrm{SL}_2(\ZZ))\) is the reduced row echelon basis. It’s a lemma that it has all integer coefficients, and a rather nice diagonal shape.
sage: victor_miller_basis(24, 6)
[
1 + 52416000*q^3 + 39007332000*q^4 + 6609020221440*q^5 + O(q^6),
q + 195660*q^3 + 12080128*q^4 + 44656110*q^5 + O(q^6),
q^2 - 48*q^3 + 1080*q^4 - 15040*q^5 + O(q^6)
]
sage: dimension_modular_forms(1,200)
17
sage: B = victor_miller_basis(200, 18) #5 seconds
sage: B
[
1 + 79288314420681734048660707200000*q^17 + O(q^18),
q + 2687602718106772837928968846869*q^17 + O(q^18),
...
q^16 + 96*q^17 + O(q^18)
]
Note: Craig Citro has made the above computation an order of magnitude faster in code he hasn’t quite got into Sage yet.
“I’ll clean those up and submit them soon, since I need them for something I’m working on … I’m currently in the process of making spaces of modular forms of level one subclass the existing code, and actually take advantage of all our fast \(E_k\) and \(\Delta\) computation code, as well as cleaning things up a bit.”